On colimits and model structures in various categories of manifolds (Q6629775)

The paper is mostly expository and aims to provide a comprehensive understanding of how model structures can be applied to different categories of manifolds. It starts by explaining the importance of model categories in abstract homotopy theory. The author provides concrete examples demonstrating that various categories of manifolds do not have all finite colimits, and hence cannot be model categories. This is significant because having all limits and colimits is an important axiom for model categories. The paper then considers various enlargements of the categories of manifolds, culminating in categories of presheaves. The author explains how to produce model structures on these enlarged categories. Finally, the paper addresses an open problem involving Poincaré spaces, providing a solution within the framework of the enlarged categories.